.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/user-guide/tutorials/02-linear_inversion/plot_inv_2_inversion_irls.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_content_user-guide_tutorials_02-linear_inversion_plot_inv_2_inversion_irls.py: Sparse Inversion with Iteratively Re-Weighted Least-Squares =========================================================== Least-squares inversion produces smooth models which may not be an accurate representation of the true model. Here we demonstrate the basics of inverting for sparse and/or blocky models. Here, we used the iteratively reweighted least-squares approach. For this tutorial, we focus on the following: - Defining the forward problem - Defining the inverse problem (data misfit, regularization, optimization) - Defining the paramters for the IRLS algorithm - Specifying directives for the inversion - Recovering a set of model parameters which explains the observations .. GENERATED FROM PYTHON SOURCE LINES 18-37 .. code-block:: Python import numpy as np import matplotlib.pyplot as plt from discretize import TensorMesh from simpeg import ( simulation, maps, data_misfit, directives, optimization, regularization, inverse_problem, inversion, ) # sphinx_gallery_thumbnail_number = 3 .. GENERATED FROM PYTHON SOURCE LINES 38-44 Defining the Model and Mapping ------------------------------ Here we generate a synthetic model and a mappig which goes from the model space to the row space of our linear operator. .. GENERATED FROM PYTHON SOURCE LINES 44-65 .. code-block:: Python nParam = 100 # Number of model paramters # A 1D mesh is used to define the row-space of the linear operator. mesh = TensorMesh([nParam]) # Creating the true model true_model = np.zeros(mesh.nC) true_model[mesh.cell_centers_x > 0.3] = 1.0 true_model[mesh.cell_centers_x > 0.45] = -0.5 true_model[mesh.cell_centers_x > 0.6] = 0 # Mapping from the model space to the row space of the linear operator model_map = maps.IdentityMap(mesh) # Plotting the true model fig = plt.figure(figsize=(8, 5)) ax = fig.add_subplot(111) ax.plot(mesh.cell_centers_x, true_model, "b-") ax.set_ylim([-2, 2]) .. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_001.png :alt: plot inv 2 inversion irls :srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none (-2.0, 2.0) .. GENERATED FROM PYTHON SOURCE LINES 66-73 Defining the Linear Operator ---------------------------- Here we define the linear operator with dimensions (nData, nParam). In practive, you may have a problem-specific linear operator which you would like to construct or load here. .. GENERATED FROM PYTHON SOURCE LINES 73-104 .. code-block:: Python # Number of data observations (rows) nData = 20 # Create the linear operator for the tutorial. The columns of the linear operator # represents a set of decaying and oscillating functions. jk = np.linspace(1.0, 60.0, nData) p = -0.25 q = 0.25 def g(k): return np.exp(p * jk[k] * mesh.cell_centers_x) * np.cos( np.pi * q * jk[k] * mesh.cell_centers_x ) G = np.empty((nData, nParam)) for i in range(nData): G[i, :] = g(i) # Plot the columns of G fig = plt.figure(figsize=(8, 5)) ax = fig.add_subplot(111) for i in range(G.shape[0]): ax.plot(G[i, :]) ax.set_title("Columns of matrix G") .. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_002.png :alt: Columns of matrix G :srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(0.5, 1.0, 'Columns of matrix G') .. GENERATED FROM PYTHON SOURCE LINES 105-111 Defining the Simulation ----------------------- The simulation defines the relationship between the model parameters and predicted data. .. GENERATED FROM PYTHON SOURCE LINES 111-115 .. code-block:: Python sim = simulation.LinearSimulation(mesh, G=G, model_map=model_map) .. GENERATED FROM PYTHON SOURCE LINES 116-122 Predict Synthetic Data ---------------------- Here, we use the true model to create synthetic data which we will subsequently invert. .. GENERATED FROM PYTHON SOURCE LINES 122-130 .. code-block:: Python # Standard deviation of Gaussian noise being added std = 0.02 np.random.seed(1) # Create a SimPEG data object data_obj = sim.make_synthetic_data(true_model, noise_floor=std, add_noise=True) .. GENERATED FROM PYTHON SOURCE LINES 131-140 Define the Inverse Problem -------------------------- The inverse problem is defined by 3 things: 1) Data Misfit: a measure of how well our recovered model explains the field data 2) Regularization: constraints placed on the recovered model and a priori information 3) Optimization: the numerical approach used to solve the inverse problem .. GENERATED FROM PYTHON SOURCE LINES 140-164 .. code-block:: Python # Define the data misfit. Here the data misfit is the L2 norm of the weighted # residual between the observed data and the data predicted for a given model. # Within the data misfit, the residual between predicted and observed data are # normalized by the data's standard deviation. dmis = data_misfit.L2DataMisfit(simulation=sim, data=data_obj) # Define the regularization (model objective function). Here, 'p' defines the # the norm of the smallness term and 'q' defines the norm of the smoothness # term. reg = regularization.Sparse(mesh, mapping=model_map) reg.reference_model = np.zeros(nParam) p = 0.0 q = 0.0 reg.norms = [p, q] # Define how the optimization problem is solved. opt = optimization.ProjectedGNCG( maxIter=100, lower=-2.0, upper=2.0, maxIterLS=20, cg_maxiter=30, cg_rtol=1e-3 ) # Here we define the inverse problem that is to be solved inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt) .. GENERATED FROM PYTHON SOURCE LINES 165-172 Define Inversion Directives --------------------------- Here we define any directiveas that are carried out during the inversion. This includes the cooling schedule for the trade-off parameter (beta), stopping criteria for the inversion and saving inversion results at each iteration. .. GENERATED FROM PYTHON SOURCE LINES 172-199 .. code-block:: Python # Add sensitivity weights but don't update at each beta sensitivity_weights = directives.UpdateSensitivityWeights(every_iteration=False) # Reach target misfit for L2 solution, then use IRLS until model stops changing. IRLS = directives.UpdateIRLS(max_irls_iterations=40, f_min_change=1e-4) # Defining a starting value for the trade-off parameter (beta) between the data # misfit and the regularization. starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=1e0) # Update the preconditionner update_Jacobi = directives.UpdatePreconditioner() # Save output at each iteration saveDict = directives.SaveOutputEveryIteration(save_txt=False) # Define the directives as a list directives_list = [ sensitivity_weights, IRLS, starting_beta, update_Jacobi, saveDict, ] .. rst-class:: sphx-glr-script-out .. code-block:: none /home/vsts/work/1/s/simpeg/directives/_directives.py:1865: FutureWarning: SaveEveryIteration.save_txt has been deprecated, please use SaveEveryIteration.on_disk. It will be removed in version 0.26.0 of SimPEG. /home/vsts/work/1/s/simpeg/directives/_directives.py:1866: FutureWarning: SaveEveryIteration.save_txt has been deprecated, please use SaveEveryIteration.on_disk. It will be removed in version 0.26.0 of SimPEG. .. GENERATED FROM PYTHON SOURCE LINES 200-206 Setting a Starting Model and Running the Inversion -------------------------------------------------- To define the inversion object, we need to define the inversion problem and the set of directives. We can then run the inversion. .. GENERATED FROM PYTHON SOURCE LINES 206-216 .. code-block:: Python # Here we combine the inverse problem and the set of directives inv = inversion.BaseInversion(inv_prob, directives_list) # Starting model starting_model = 1e-4 * np.ones(nParam) # Run inversion recovered_model = inv.run(starting_model) .. rst-class:: sphx-glr-script-out .. code-block:: none Running inversion with SimPEG v0.25.1 ================================================= Projected GNCG ================================================= # beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment ----------------------------------------------------------------------------------------------------------------- 0 1.71e+06 3.75e+03 1.03e-09 3.75e+03 0 inf inf 1 1.71e+06 1.94e+03 3.75e-04 2.58e+03 1.96e+01 0 8 4.37e-04 2.14e+00 2 8.54e+05 1.35e+03 8.73e-04 2.09e+03 1.91e+01 0 9 2.35e-04 1.93e-01 3 4.27e+05 8.09e+02 1.77e-03 1.56e+03 1.86e+01 0 9 8.58e-04 5.18e-01 4 2.13e+05 4.15e+02 3.07e-03 1.07e+03 1.76e+01 0 10 8.54e-04 3.62e-01 5 1.07e+05 1.85e+02 4.57e-03 6.73e+02 1.54e+01 0 12 9.85e-04 2.72e-01 6 5.33e+04 7.52e+01 5.99e-03 3.94e+02 1.34e+01 0 16 5.21e-04 8.72e-02 7 2.67e+04 3.09e+01 7.12e-03 2.21e+02 1.17e+01 0 18 8.88e-04 8.46e-02 8 1.33e+04 1.48e+01 7.94e-03 1.21e+02 9.47e+00 0 28 8.33e-04 4.32e-02 Reached starting chifact with l2-norm regularization: Start IRLS steps... irls_threshold 1.2644411719065778 9 1.33e+04 2.38e+01 9.74e-03 1.54e+02 1.45e+01 0 30 1.16e-03 3.74e-02 10 1.06e+04 2.49e+01 1.12e-02 1.43e+02 6.02e+00 0 30 1.30e-03 1.40e-02 11 8.24e+03 2.45e+01 1.24e-02 1.26e+02 2.70e+00 0 29 9.42e-04 9.01e-03 12 6.46e+03 2.35e+01 1.34e-02 1.10e+02 3.78e+00 0 29 7.11e-04 6.01e-03 13 5.18e+03 2.19e+01 1.40e-02 9.44e+01 4.25e+00 0 28 7.70e-04 6.23e-03 14 5.18e+03 2.28e+01 1.35e-02 9.28e+01 7.32e+00 0 26 9.75e-04 7.46e-03 15 4.21e+03 2.03e+01 1.35e-02 7.70e+01 4.46e+00 0 22 7.35e-04 6.71e-03 16 4.21e+03 1.97e+01 1.23e-02 7.14e+01 6.24e+00 0 23 9.25e-04 6.95e-03 17 4.21e+03 1.93e+01 1.11e-02 6.62e+01 6.64e+00 0 23 2.80e-04 2.14e-03 18 4.21e+03 1.89e+01 9.98e-03 6.09e+01 6.65e+00 0 22 8.85e-04 6.83e-03 19 4.21e+03 1.84e+01 8.84e-03 5.56e+01 6.62e+00 0 21 2.80e-04 2.21e-03 20 4.21e+03 1.78e+01 7.74e-03 5.03e+01 6.96e+00 0 19 9.55e-04 7.73e-03 21 6.58e+03 2.08e+01 5.93e-03 5.98e+01 1.43e+01 0 18 9.45e-04 2.94e-02 22 6.58e+03 2.06e+01 5.24e-03 5.51e+01 9.83e+00 0 19 9.33e-04 1.40e-02 23 6.58e+03 1.93e+01 4.49e-03 4.89e+01 9.77e+00 0 23 2.80e-04 4.27e-03 24 6.58e+03 1.76e+01 3.82e-03 4.27e+01 1.00e+01 0 25 2.73e-04 3.42e-03 25 1.03e+04 1.91e+01 2.88e-03 4.89e+01 1.50e+01 0 18 6.82e-04 3.12e-02 26 1.03e+04 1.86e+01 2.54e-03 4.48e+01 1.06e+01 0 21 7.35e-04 1.05e-02 27 1.03e+04 1.80e+01 2.30e-03 4.18e+01 1.18e+01 0 21 8.23e-04 1.87e-02 28 1.03e+04 1.72e+01 2.08e-03 3.86e+01 1.20e+01 0 23 7.84e-04 2.13e-02 29 1.63e+04 1.87e+01 1.67e-03 4.59e+01 1.72e+01 0 19 9.30e-04 9.14e-02 30 1.63e+04 1.78e+01 1.42e-03 4.09e+01 1.03e+01 0 20 6.29e-04 2.24e-02 31 2.55e+04 1.90e+01 1.12e-03 4.74e+01 1.74e+01 0 22 4.38e-04 5.78e-02 32 2.55e+04 1.81e+01 9.30e-04 4.18e+01 9.97e+00 0 23 5.06e-04 2.63e-02 33 2.55e+04 1.71e+01 8.05e-04 3.77e+01 1.18e+01 0 25 9.77e-04 4.53e-02 34 4.04e+04 1.81e+01 6.49e-04 4.43e+01 1.80e+01 0 24 8.58e-04 1.47e-01 35 4.04e+04 1.74e+01 5.45e-04 3.94e+01 1.03e+01 0 26 6.38e-04 2.06e-02 36 6.36e+04 1.84e+01 4.37e-04 4.61e+01 1.78e+01 0 23 9.49e-04 1.92e-01 37 6.36e+04 1.76e+01 3.64e-04 4.07e+01 1.04e+01 0 24 9.01e-04 4.32e-02 38 9.97e+04 1.84e+01 2.91e-04 4.74e+01 1.76e+01 0 24 6.52e-04 1.99e-01 39 9.97e+04 1.76e+01 2.41e-04 4.17e+01 1.05e+01 0 24 7.37e-04 1.28e-01 40 1.56e+05 1.85e+01 1.93e-04 4.87e+01 1.81e+01 0 24 3.57e-04 1.52e-01 41 1.56e+05 1.78e+01 1.60e-04 4.28e+01 1.09e+01 0 26 7.45e-04 2.91e-02 42 2.44e+05 1.86e+01 1.29e-04 5.03e+01 1.83e+01 0 26 3.12e-04 7.75e-02 43 2.44e+05 1.80e+01 1.07e-04 4.42e+01 1.12e+01 0 27 4.42e-04 2.16e-02 44 3.80e+05 1.88e+01 8.71e-05 5.19e+01 1.84e+01 0 26 7.59e-04 1.92e-01 45 3.80e+05 1.83e+01 7.34e-05 4.61e+01 1.19e+01 0 27 5.35e-04 2.63e-02 46 3.80e+05 1.76e+01 6.29e-05 4.15e+01 1.33e+01 0 29 6.79e-04 4.64e-02 47 5.96e+05 1.83e+01 5.10e-05 4.87e+01 1.87e+01 0 26 6.91e-04 1.84e-01 48 5.96e+05 1.80e+01 4.41e-05 4.43e+01 1.27e+01 0 27 9.66e-04 7.00e-02 Reach maximum number of IRLS cycles: 40 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 8.1827e-01 <= tolF*(1+|f0|) = 3.7528e+02 1 : |xc-x_last| = 7.0829e-02 <= tolX*(1+|x0|) = 1.0010e-01 0 : |proj(x-g)-x| = 1.2730e+01 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.2730e+01 <= 1e3*eps = 1.0000e-02 0 : maxIter = 100 <= iter = 48 ------------------------- DONE! ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 217-220 Plotting Results ---------------- .. GENERATED FROM PYTHON SOURCE LINES 220-261 .. code-block:: Python fig, ax = plt.subplots(1, 2, figsize=(12 * 1.2, 4 * 1.2)) # True versus recovered model ax[0].plot(mesh.cell_centers_x, true_model, "k-") ax[0].plot(mesh.cell_centers_x, inv_prob.l2model, "b-") ax[0].plot(mesh.cell_centers_x, recovered_model, "r-") ax[0].legend(("True Model", "Recovered L2 Model", "Recovered Sparse Model")) ax[0].set_ylim([-2, 2]) # Observed versus predicted data ax[1].plot(data_obj.dobs, "k-") ax[1].plot(inv_prob.dpred, "ko") ax[1].legend(("Observed Data", "Predicted Data")) # Plot convergence fig = plt.figure(figsize=(9, 5)) ax = fig.add_axes([0.2, 0.1, 0.7, 0.85]) ax.plot(saveDict.phi_d, "k", lw=2) twin = ax.twinx() twin.plot(saveDict.phi_m, "k--", lw=2) ax.plot( np.r_[IRLS.metrics.start_irls_iter, IRLS.metrics.start_irls_iter], np.r_[0, np.max(saveDict.phi_d)], "k:", ) ax.text( IRLS.metrics.start_irls_iter, 0.0, "IRLS Start", va="bottom", ha="center", rotation="vertical", size=12, bbox={"facecolor": "white"}, ) ax.set_ylabel(r"$\phi_d$", size=16, rotation=0) ax.set_xlabel("Iterations", size=14) twin.set_ylabel(r"$\phi_m$", size=16, rotation=0) .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_003.png :alt: plot inv 2 inversion irls :srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_003.png :class: sphx-glr-multi-img * .. image-sg:: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_004.png :alt: plot inv 2 inversion irls :srcset: /content/user-guide/tutorials/02-linear_inversion/images/sphx_glr_plot_inv_2_inversion_irls_004.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(865.1527777777777, 0.5, '$\\phi_m$') .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 30.986 seconds) **Estimated memory usage:** 321 MB .. _sphx_glr_download_content_user-guide_tutorials_02-linear_inversion_plot_inv_2_inversion_irls.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_inv_2_inversion_irls.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_2_inversion_irls.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_inv_2_inversion_irls.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_